Concern for achieving unity "involves the whole Church, faithful and clergy alike." But we must realize "that this holy objective - the reconciliation of all Christians in the unity of the one and only Church of Christ - transcends human powers and gifts." That is why we place all our hope "in the prayer of Christ for the Church, in the love of the Father for us, and in the power of the Holy Spirit." CCC 822
In the churches and ecclesial communities which are separated from full communion with the Catholic Church, many elements of sanctification and truth can be found. All of these blessings come from Christ and lead to Catholic unity. Members of these churches and communities are incorporated into Christ by Baptism and so we recognize them as brothers. (#163)
I am a simple layman and have no idea how these schisms can be healed. I don't believe there is anything man can do on his own, but it will take His grace and the working of the Holy Spirit to overcome what our mutual sins, pride, and stubbornness have brought about. For now, all I can see from my perspective is we need to rachet down the rhetoric as much as possible (this doesn't mean not defending the truths of the faith), inter-communal dialogue among theologians, working together whenever possible (e.g. pro-life causes, charity, etc.), and getting to know and respect one another as much as we can instead of relying on old stereotypes. Christ never wanted this and our divisions are not of His doing, but our own. I like how the Catechism Companion, Vol I puts this:
Sin is what causes division among us. At the Last Supper, Jesus begs his Father that his followers may all be one. We should never be content with division in the Body of Christ when Jesus specifically prayed for us to be united. This should cause us great pain; if we love what Jesus loves, then we should love unity and be hurt by its absence... The truth of the Catholic Church is a gift. Like St. Paul, we should not boast in anything except for the cross of Jesus Christ. (p. 234)